We prove various bounds on moments

where fm is a solution of the discrete (respectively, continuous) Smoluchowski coagulation–fragmentation equations with diffusion. In a previous paper we proved similar results for all weak solutions to the discrete Smoluchowski equation provided that there is no fragmentation and certain moments are bounded in suitable Lq-spaces initially. We prove the corresponding results in the case of the continuous Smoluchowski equation. When there is also fragmentation, we need to assume that the solution f is regular in the sense that f can be approximated by solutions to the Smoluchowski equation for which the coagulation and fragmentation coefficients are 0 when the cluster sizes are large. We also need suitable assumptions on the coagulation rates to avoid gelation. On the fragmentation rate β we assume that supn supm≤l β(m, n)/n < ∞ for every positive l, and that there exist constants a0 ≥ 0 and c0 such that β(n,m) ≤ c0(n + m)a0a0.

We study the Bhatnagar–Gross–Krook (BGK) approximation to first-order scalar conservation laws with a flux which is discontinuous in the space variable. We show that the Cauchy problem for the BGK approximation is well posed and that, as the relaxation parameter tends to 0, it converges to the (entropy) solution of the limit problem.

We consider a non-autonomous competitive model with generalized functional responses for interaction among n species, the adult members of which are in competition. For each of the n species the model incorporates a distributed time delay which represents the time from birth to maturity of that species. Based on some comparison arguments, we discuss the permanence and extinction of the species. By virtue of the continuation theorem of coincidence degree theory, we prove the existence of a positive periodic solution. By means of constructing appropriate Lyapunov functionals, we obtain sufficient conditions for the uniqueness and the global stability of the periodic solution. Two examples are given to illustrate the feasibility of our main results.

The semi-classical regime of standing wave solutions of a Schrödinger equation in the presence of non-constant electric and magnetic potentials is studied in the case of non-local nonlinearities of Hartree type. It is shown that there exists a family of solutions having multiple concentration regions which are located around the minimum points of the electric potential.

We study the orbital stability of standing waves for the Klein–Gordon–Schrödinger system in two spatial dimensions. It is proved that the standing wave is stable if the frequency is sufficiently small. To prove this, we obtain the uniqueness of ground state and investigate the spectrum of the appropriate linearized operator by using the perturbation method developed by Genoud and Stuart and Lin and Wei. Then we apply to our system the general theory of Grillakis, Shatah and Strauss.

Let Hn be a monotone sequence of non-negative self-adjoint operators or relations in a Hilbert space. Then there exists a self-adjoint relation H∞ such that Hn converges to H∞ in the strong resolvent sense. This result and related limit results are explored in detail and new simple proofs are presented. The corresponding statements for monotone sequences of semi-bounded closed forms are established as immediate consequences. Applications and examples, illustrating the general results, include sequences of multiplication operators, Sturm–Liouville operators with increasing potentials, forms associated with Kreĭn–Feller differential operators, singular perturbations of non-negative self-adjoint operators and the characterization of the Friedrichs and Kreĭn–von Neumann extensions of a non-negative operator or relation.

AC rings (rings satisfying Auslander's condition) have recently been studied by Christensen and Holm, among others. We will employ tilting theory to study Auslander's condition. In particular, we prove that the class of AC Artin algebras is closed under tilting equivalences.

We study the homogenization and localization of high-frequency waves in a locally periodic media with period ε. We consider initial data that are localized Bloch-wave packets, i.e. that are the product of a fast oscillating Bloch wave at a given frequency ξ and of a smooth envelope function whose support is concentrated at a point x with length scale . We assume that (ξ, x) is a stationary point in the phase space of the Hamiltonian λ(ξ, x), i.e. of the corresponding Bloch eigenvalue. Upon rescaling at size we prove that the solution of the wave equation is approximately the sum of two terms with opposite phases which are the product of the oscillating Bloch wave and of two limit envelope functions which are the solution of two Schrödinger type equations with quadratic potential. Furthermore, if the full Hessian of the Hamiltonian λ(ξ, x) is positive definite, then localization takes place in the sense that the spectrum of each homogenized Schrödinger equation is made of a countable sequence of finite multiplicity eigenvalues with exponentially decaying eigenfunctions.

We introduce a polynomial invariant of graphs on surfaces, PG, generalizing the classical Tutte polynomial. Topological duality on surfaces gives rise to a natural duality result for PG, analogous to the duality for the Tutte polynomial of planar graphs. This property is important from the perspective of statistical mechanics, where the Tutte polynomial is known as the partition function of the Potts model. For ribbon graphs, PG specializes to the well-known Bollobás–Riordan polynomial, and in fact the two polynomials carry equivalent information in this context. Duality is also established for a multivariate version of the polynomial PG. We then consider a 2-variable version of the Jones polynomial for links in thickened surfaces, taking into account homological information on the surface. An analogue of Thistlethwaite's theorem is established for these generalized Jones and Tutte polynomials for virtual links.

We prove that the Calabi–Yau equation can be solved on the Kodaira–Thurston manifold for all given T2-invariant volume forms. This provides support for Donaldson's conjecture that Yau's theorem has an extension to symplectic 4-manifolds with compatible but non-integrable almost complex structures.

Let Ωn be the nn-element set consisting of all functions that have {1, 2, 3, . . ., n} as both domain and codomain. Let T(f) be the order of f, i.e., the period of the sequence f, f(2), f(3), f(4) . . . of compositional iterates. A closely related number, B(f) = the product of the lengths of the cycles of f, has previously been used as an approximation for T. This paper proves that the average values of these two quantities are quite different. The expected value of T iswhere k0 is a complicated but explicitly defined constant that is approximately 3.36. The expected value of B is much larger:

Let k ≥ 1 be an integer, and let H be a graph with no isolated vertices embedded in the projective plane, such that every homotopically non-trivial closed curve intersects H at least k times, and the deletion and contraction of any edge in this embedding results in an embedding that no longer has this property. Let G be the planar double cover of H obtained by lifting G into the universal covering space of the projective plane, the sphere. We prove that G is minor-minimal of branch-width 2k. We also exhibit examples of minor-minimal planar graphs of branch-width 6 that do not arise in this way.

We investigate Tukey functions from the ideal of all closed nowhere-dense subsets of 2ℕ. In particular, we answer an old question of Isbell and Fremlin by showing that this ideal is not Tukey reducible to the ideal of density zero subsets of ℕ. We also prove non-existence of various special types of Tukey reductions from the nowhere-dense ideal to analytic P-ideals. In connection with these results, we study families of clopen subsets of 2ℕ with the property that for each nowhere-dense subset of 2ℕ there is a set in not intersecting it. We call such families avoiding.

Soit p un nombre premier et F un corps local non archimédien de caractéristique p. Dans cet article, à une représentation lisse irréductible de GL2(F) sur avec caractère central, nous associons un diagramme qui détermine la représentation de départ à isomorphisme près. Nous le déterminons également dans certains cas.

We compute the Grothendieck and Picard groups of a smooth toric DM stack by using a suitable category of graded modules over a polynomial ring. The polynomial ring with a suitable grading and suitable irrelevant ideal functions is a homogeneous coordinate ring for the stack.

Let be a lattice in the real simple Lie group L. If L is of rank at least 2 (respectively locally isomorphic to Sp(n, 1)) any unbounded morphism ρ : Γ → G into a simple real Lie group G essentially extends to a Lie morphism ρL : L → G (Margulis's superrigidity theorem, respectively Corlette's theorem). In particular any such morphism is infinitesimally, thus locally, rigid. On the other hand, for L = SU(n, 1) even morphisms of the form are not infinitesimally rigid in general. Almost nothing is known about their local rigidity. In this paper we prove that any cocompact lattice Γ in SU(n, 1) is essentially locally rigid (while in general not infinitesimally rigid) in the quaternionic groups Sp(n, 1), SU(2n, 2) or SO(4n, 4) (for the natural sequence of embeddings SU(n, 1) ⊂ Sp(n, 1) ⊂ SU(2n, 2) ⊂ SO(4n, 4)).

Walter Wilson Stothers was born in Glasgow on 8 November 1946. A third (youngest) son, he had the identical name to his father. From childhood, however, he had always been known by his middle name ‘Wilson’, so that his father, a Glasgow GP, would never be referred to as ‘Old Walter’. His mother, as Jean Young Kyle, had herself graduated in Mathematics in 1927, a rare achievement for a woman at that time. After attending the local primary school 1952–1956, Wilson completed his primary education in the preparatory classes in Allan Glen's School, then a distinguished Glasgow boys school with a scientific emphasis, progressing to the secondary school in 1958 and ending by becoming Dux in 1964. He also played in the school rugby first XV. From 1964–1968 he was a student in the Science Faculty of Glasgow University. His original intention was to take Honours in Chemistry and, indeed, he won the Chemistry prize in his first year. But he excelled in all subjects, winning the Faraday medal in the Intermediate Honours (second year) class in Natural Philosophy (Physics). After that he concentrated on Mathematics and became the top student, gaining a First Class Honours degree, as well as the Cunninghame Medal and a Jack Scholarship to Peterhouse College, Cambridge (1968). Before commencing postgraduate studies he married Andrea Watson in September 1968. At Cambridge from 1968–1971 he studied for a Ph.D. in number theory under the supervision of Peter Swinnerton-Dyer and graduated in 1972 with the thesis Some Discrete Triangle Groups. By then he was becoming aware of the strange realm inhabited by mathematicians that he seemed to be entering. So when his Cambridge room-mate Bob Odoni, at a research meeting they were attending as postgrads, asked, ‘Wilson, do you realise that we are the only normal people here’, Wilson felt compelled to respond, ‘What makes you think that we are normal?’